Outlier Detection for Multivariate Time Series Using Dynamic Bayesian Networks
<p>Example of a stationary first-order Markov DBN. On the left, the prior network <math display="inline"><semantics> <msup> <mi>B</mi> <mn>0</mn> </msup> </semantics></math>, for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, and on the right, the transition network <math display="inline"><semantics> <msubsup> <mi>B</mi> <mrow> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>t</mi> </msubsup> </semantics></math> over slices <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> and <span class="html-italic">t</span>, for all <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>≥</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 2
<p>Scheme of the proposed outlier detection approach comprised of four phases. Datasets formed by MTS data can be directly applied to the modeling phase when discrete; otherwise, the pre-processing phase is applied before modeling. Discrete data is delivered to the modeling phase along with parameters <span class="html-italic">p</span>, <span class="html-italic">m</span>, and <span class="html-italic">s</span> of the DBN to be modeled. Afterward, a sliding window algorithm outputs a score distribution for the data (scoring entire MTS, called subjects, or only portions of it, called transitions, depending on the user’s choice). The score-analysis phase considers two distinct strategies providing thus two possible routes for outlier disclosure.</p> "> Figure 3
<p>Transition networks of stationary first-order DBNs (<math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>). The network (<b>a</b>) on the left represents the transition network of DBN <span class="html-italic">A</span> which generates normal subjects. Networks (<b>b</b>,<b>c</b>) represent DBN <span class="html-italic">B</span> and <span class="html-italic">C</span>, respectively, which generate anomalous subjects. Dashed connections represent links which are removed with respect to the normal network (<b>a</b>), while red links symbolize added dependencies. Solid black edges are connections which are common with respect to (<b>a</b>).</p> "> Figure 4
<p>Comparison between GMM and Tukey’s score-analysis <math display="inline"><semantics> <msub> <mi mathvariant="normal">F</mi> <mn>1</mn> </msub> </semantics></math> scores for multiple outlier ratios. Each value is an average of all 15 trials performed for each outlier ratio.</p> "> Figure 5
<p>Subject outlierness using METEOR (<b>a</b>) and PST approach (<b>b</b>) for a same experiment of a dataset of 10,000 subjects (<math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>10</mn> <mo>,</mo> <mn>000</mn> </mrow> </semantics></math>) with 20% anomalies generated by model <span class="html-italic">C</span>. Histograms display thresholds using both score-analysis strategies. Scores below the threshold are classified as abnormal (in red) while the rest are classified as normal (in green), being the presented color representation for the Tukey’s thresholds.</p> "> Figure 6
<p>Mean and standard deviation of normalized ECG variables along time using a SAX alphabet <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </semantics></math>.</p> "> Figure 7
<p>ECG transitions arranged by subject. A non-stationary second-order tDBN (<math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>) model with inter-slice connectivity (<math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>) is used together with Tukey’s score-analysis. Flipped subjects are associated to the highest subject ids. Data is discretized using SAX with an alphabet of 5 symbols (<math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> for all <span class="html-italic">i</span>). Transitions displayed in red are classified as abnormal while in green are classified as normal.</p> "> Figure 8
<p>Normalized values of variables <math display="inline"><semantics> <msub> <mi>X</mi> <mrow> <mi>i</mi> <mo>∈</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>6</mn> </mrow> </msub> </semantics></math> representing France’s mortality rates of males with ages 10, 20, 30, 40, 60 and 80, respectively, from 1841 to 1987. Each time stamp represents a year. Data is discretized with a SAX alphabet <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> for all <span class="html-italic">i</span>.</p> "> Figure 9
<p>Transition outlierness for mortality datasets of 5 (<b>a</b>) and 6 (<b>b</b>) variables using a third-order tDBN (<math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>) with one inter-slice connectivity per node (<math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>). Dataset (<b>a</b>) is comprised by 5 variables (<math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>) representing mortality rates of males with ages 20, 30, 40, 60 and 80. Dataset (<b>b</b>) includes the same variables as (<b>a</b>) with the addition of a variable representing the mortality rate of males aged 10 (<math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>). Transitions are arranged by year and classified as anomalous (red) and normal (green). Major wars and epidemics which affected France in the selected years are exhibited.</p> ">
Abstract
:1. Introduction
2. Theoretical Background
2.1. Bayesian Networks
2.2. Dynamic Bayesian Networks
3. Methods
3.1. Pre-Processing
Algorithm 1 Data Pre-Processing |
Input: A MTS dataset D of n variables along T instants; an alphabet size for each attribute , ; desired length of the resulting MTS. Output: The set of input MTS discretized. 1: procedure SAX(D, for all i,w) 2: for each subject h in D do 3: for each TS , with do 4: for each t, with do 5: Normhihi[t] ▹Normalization 6: function PAA() ▹Dimensionality reduction 7: 8: Partition the in contiguous blocks of size 9: for each block do 10: ▹ Compressed slices 11: 12: function Discretization () ▹ Symbolic discretization 13: ) 14: for each value in do 15: Discretehi[k] 16: ▹ Return discretized MTS dataset |
3.2. Modeling
Algorithm 2 Optimal Non-Stationary m-Order Markov tDBN Learning |
Input: A set of input MTS discretized over w time slices; the Markov lag m; the maximum number of parents p from preceding time slices. Output: A tree-augmented DBN structure. 1: procedure Tree-augmented DBN(MTS,m,p) 2: for each transition do 3: Build a complete directed graph in 4: Calculate the weight of all edges and the optimal set of parents 5: Apply a maximum branching algorithm 6: Extract transition network and the optimal set of parents 7: Collect transition networks to obtain a tDBN structure |
3.3. Scoring
Algorithm 3 Transition Outlier Detection |
Input: A tDBN storing conditional probabilities for each transition network , a (discretized) MTS dataset D, and a threshold to discern abnormality. Output: The set of anomalous transitions with scores below . 1: procedure 2: for each time slice t do 3: for each subject do 4: function Scoring() 5: for each variable do 6: ΠXi[t] 7: whi[t] 8: phi 9: Phi ▹ Probability smoothing 10: sht−m:t ▹ Transition score 11: if then 12: outliers .append |
3.4. Parameter Tuning
3.5. Score-Analysis
3.5.1. Tukey’s Strategy
3.5.2. Gaussian Mixture Model
4. Experimental Results
4.1. Simulated Data
4.1.1. Tukey’s Score-Analysis
4.1.2. Gaussian Mixture Model
4.1.3. Comparison between GMM and Tukey’s Score-Analysis
4.1.4. Comparison with Probabilistic Suffix Trees
4.2. ECG
4.3. Mortality
4.4. Pen-Digits
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Model B | Model C | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
PPV | TPR | ACC | PPV | TPR | ACC | |||||
100 | 0.88 | 0.70 | 0.98 | 0.78 | 100 | 0.89 | 0.73 | 0.98 | 0.80 | |
5 | 1000 | 0.93 | 0.96 | 0.99 | 0.94 | 1000 | 0.91 | 0.98 | 0.99 | 0.94 |
10,000 | 0.95 | 0.98 | 0.99 | 0.96 | 10,000 | 0.94 | 1.00 | 0.99 | 0.97 | |
100 | 0.96 | 0.38 | 0.94 | 0.54 | 100 | 0.89 | 0.73 | 0.97 | 0.80 | |
10 | 1000 | 0.99 | 0.87 | 0.99 | 0.93 | 1000 | 0.97 | 0.87 | 0.98 | 0.92 |
10,000 | 0.99 | 0.91 | 0.99 | 0.95 | 10,000 | 0.99 | 0.87 | 0.98 | 0.93 | |
100 | 1.00 | 0.19 | 0.83 | 0.32 | 100 | 0.90 | 0.22 | 0.84 | 0.35 | |
20 | 1000 | 1.00 | 0.20 | 0.84 | 0.33 | 1000 | 1.00 | 0.37 | 0.87 | 0.54 |
10,000 | 1.00 | 0.16 | 0.83 | 0.28 | 10,000 | 1.00 | 0.29 | 0.86 | 0.45 |
Model B | Model C | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
PPV | TPR | ACC | PPV | TPR | ACC | |||||
100 | 0.82 | 0.70 | 0.98 | 0.76 | 100 | 0.64 | 1.00 | 0.96 | 0.78 | |
5 | 1000 | 0.91 | 0.97 | 0.99 | 0.94 | 1000 | 0.86 | 0.99 | 0.99 | 0.92 |
10,000 | 0.95 | 0.98 | 0.99 | 0.96 | 10,000 | 0.98 | 1.00 | 0.99 | 0.99 | |
100 | 0.77 | 0.68 | 0.93 | 0.72 | 100 | 0.92 | 0.78 | 0.97 | 0.84 | |
10 | 1000 | 0.94 | 0.96 | 0.99 | 0.95 | 1000 | 0.89 | 0.97 | 0.98 | 0.93 |
10,000 | 0.91 | 0.98 | 0.99 | 0.94 | 10,000 | 0.93 | 0.96 | 0.99 | 0.95 | |
100 | 0.66 | 0.49 | 0.85 | 0.56 | 100 | 0.75 | 0.58 | 0.88 | 0.65 | |
20 | 1000 | 0.86 | 0.89 | 0.94 | 0.87 | 1000 | 0.91 | 0.92 | 0.96 | 0.92 |
10,000 | 0.86 | 0.94 | 0.96 | 0.90 | 10,000 | 0.93 | 0.94 | 0.97 | 0.94 |
Tukey’s Strategy | ||||||||
---|---|---|---|---|---|---|---|---|
Model B | Model C | |||||||
PPV | TPR | ACC | PPV | TPR | ACC | |||
5 | 0.96 | 0.73 | 0.98 | 0.83 | 0.96 | 0.94 | 0.99 | 0.95 |
10 | 0.70 | 0.02 | 0.90 | 0.04 | 0.98 | 0.39 | 0.94 | 0.56 |
20 | 0.42 | 0.00 | 0.80 | 0.00 | 1.00 | 0.03 | 0.81 | 0.06 |
GMM Strategy | ||||||||
Model B | Model C | |||||||
PPV | TPR | ACC | PPV | TPR | ACC | |||
5 | 0.86 | 0.88 | 0.99 | 0.87 | 0.94 | 0.95 | 0.99 | 0.94 |
10 | 0.20 | 0.87 | 0.65 | 0.33 | 0.88 | 0.68 | 0.96 | 0.77 |
20 | 0.25 | 0.67 | 0.53 | 0.36 | 0.763 | 0.883 | 0.92 | 0.82 |
Experiment | TP | FP | TN | FN | PPV | TPR | ACC | |
---|---|---|---|---|---|---|---|---|
24 | 41 | 1102 | 106 | 0.37 | 0.18 | 0.88 | 0.25 | |
98 | 45 | 1098 | 32 | 0.69 | 0.75 | 0.94 | 0.72 | |
90 | 42 | 1101 | 40 | 0.68 | 0.69 | 0.94 | 0.69 |
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Serras, J.L.; Vinga, S.; Carvalho, A.M. Outlier Detection for Multivariate Time Series Using Dynamic Bayesian Networks. Appl. Sci. 2021, 11, 1955. https://doi.org/10.3390/app11041955
Serras JL, Vinga S, Carvalho AM. Outlier Detection for Multivariate Time Series Using Dynamic Bayesian Networks. Applied Sciences. 2021; 11(4):1955. https://doi.org/10.3390/app11041955
Chicago/Turabian StyleSerras, Jorge L., Susana Vinga, and Alexandra M. Carvalho. 2021. "Outlier Detection for Multivariate Time Series Using Dynamic Bayesian Networks" Applied Sciences 11, no. 4: 1955. https://doi.org/10.3390/app11041955